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Accounting for the Error due to Unresolved Scales in Ensemble Data Assimilation: A Comparison of Different Approaches
 3132 MONTHLY WEATHER REVIEW VOLUME
, 2005
"... Insufficient model resolution is one source of model error in numerical weather predictions. Methods for parameterizing this error in ensemble data assimilations are explored here. Experiments were conducted with a twolayer primitive equation model, where the assumed true state was a T127 forecast ..."
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Cited by 42 (4 self)
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Insufficient model resolution is one source of model error in numerical weather predictions. Methods for parameterizing this error in ensemble data assimilations are explored here. Experiments were conducted with a twolayer primitive equation model, where the assumed true state was a T127 forecast simulation. Ensemble data assimilations were performed with the same model at T31 resolution, assimilating imperfect observations drawn from the T127 forecast. By design, the magnitude of errors due to model truncation was much larger than the error growth due to initial condition uncertainty, making this a stringent test of the ability of an ensemblebased data assimilation to deal with model error. Two general methods, “covariance inflation ” and “additive error, ” were considered for parameterizing the model error at the resolved scales (T31 and larger) due to interaction with the unresolved scales (T32 to T127). Covariance inflation expanded the background forecast members ’ deviations about the ensemble mean, while additive error added specially structured noise to each ensemble member forecast before the update step. The method of parameterizing this model error had a substantial effect on the accuracy of the ensemble data assimilation. Covariance inflation produced ensembles with analysis errors that were no lower than the analysis errors from threedimensional variational (3DVar) assimilation, and for the method to avoid filter
Ensemblebased atmospheric data assimilation
, 2004
"... Ensemblebased data assimilation techniques are being explored as possible alternatives to current operational analysis techniques such as 3 or 4dimensional variational assimilation. Ensemblebased assimilation techniques utilize an ensemble of parallel data assimilation and forecast cycles. The ..."
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Cited by 42 (2 self)
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Ensemblebased data assimilation techniques are being explored as possible alternatives to current operational analysis techniques such as 3 or 4dimensional variational assimilation. Ensemblebased assimilation techniques utilize an ensemble of parallel data assimilation and forecast cycles. The backgrounderror covariances are estimated using the forecast ensemble and are used to produce an ensemble of analyses. The backgrounderror covariances are flow dependent and often have very complicated structure, providing a very different adjustment to the observations than are seen from methods such as 3 dimensional variational assimilation. Though computationally expensive, ensemblebased techniques are relatively easy to code, since no adjoint nor tangentlinear models are required, and previous tests in simple models suggest that dramatic improvements over existing operational methods may be possible. A review of the ensemblebased assimilation is provided here, starting from the basic concepts of Bayesian assimilation. Without some simplification, full Bayesian assimilation is computationally impossible for model states of large dimension. Assuming normality of error statistics and linearity of error growth, the state and its error covariance may be predicted optimally using Kalman filter (KF) techniques. The ensemble Kalman filter (EnKF) is then described. The EnKF is an approximation to the KF in that backgrounderror covariances are estimated from a finite ensemble of forecasts. However, no assumptions about linearity of error growth are made. Recent algorithmic variants on the standard EnKF are also described, as well as methods for simplifying the computations and increasing the accuracy. Examples of ensemblebased assimilations are provided in simple and more realistic dynamical systems.
Local ensemble Kalman filtering in the presence of model bias
 TELLUS 58A
, 2006
"... We modify the local ensemble Kalman filter (LEKF) to incorporate the effect of forecast model bias. The method is based on augmentation of the atmospheric state by estimates of the model bias, and we consider different ways of modeling (i.e. parameterizing) the model bias. We evaluate the effectiven ..."
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Cited by 28 (7 self)
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We modify the local ensemble Kalman filter (LEKF) to incorporate the effect of forecast model bias. The method is based on augmentation of the atmospheric state by estimates of the model bias, and we consider different ways of modeling (i.e. parameterizing) the model bias. We evaluate the effectiveness of the proposed augmented state ensemble Kalman filter through numerical experiments incorporating various model biases into the model of Lorenz and Emanuel. Our results highlight the critical role played by the selection of a good parameterization model for representing the form of the possible bias in the forecast model. In particular, we find that forecasts can be greatly improved provided that a good model parameterizing the model bias is used to augment the state in the Kalman filter.
A comparison of hybrid ensemble transform Kalman filterOptimum Interpolation and ensemble squareroot filter analysis schemes
, 2007
"... A hybrid ensemble transform Kalman filter (ETKF)–optimum interpolation (OI) analysis scheme is described and compared with an ensemble square root filter (EnSRF) analysis scheme. A twolayer primitive equation model was used under perfectmodel assumptions. A simplified observation network was used ..."
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Cited by 20 (6 self)
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A hybrid ensemble transform Kalman filter (ETKF)–optimum interpolation (OI) analysis scheme is described and compared with an ensemble square root filter (EnSRF) analysis scheme. A twolayer primitive equation model was used under perfectmodel assumptions. A simplified observation network was used, and the OI method utilized a static background error covariance constructed from a large inventory of historical forecast errors. The hybrid scheme updated the ensemble mean using a hybridized ensemble and static backgrounderror covariance. The ensemble perturbations in the hybrid scheme were updated by the ETKF scheme. The EnSRF ran parallel data assimilation cycles for each member and serially assimilated the observations. The EnSRF backgrounderror covariance was estimated fully from the ensemble. For 50member ensembles, the analyses from the hybrid scheme were as accurate or nearly as accurate as those from the EnSRF, depending on the norm. For 20member ensembles, the analyses from the hybrid scheme were more accurate than analyses from the EnSRF under certain norms. Both hybrid and EnSRF analyses were more accurate than the analyses from the OI. Further reducing the ensemble size to five members, the EnSRF exhibited filter divergence, whereas the analyses from the hybrid scheme were still better than those updated by the OI. Additionally, the hybrid scheme was less prone to spurious gravity
EnsembleBased Simultaneous State and Parameter Estimation in a TwoDimensional SeaBreeze Model
, 2005
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Ensemble transform Kalman filterbased ensemble perturbations in an operational global prediction system at NCEP, Tellus 58A
, 2006
"... The initial perturbations used for the operational global ensemble prediction system of the National Centers for Environmental Prediction are generated through the breeding method with a regional rescaling mechanism. Limitations of the system include the use of a climatologically fixed estimate of ..."
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Cited by 10 (1 self)
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The initial perturbations used for the operational global ensemble prediction system of the National Centers for Environmental Prediction are generated through the breeding method with a regional rescaling mechanism. Limitations of the system include the use of a climatologically fixed estimate of the analysis error variance and the lack of an orthogonalization in the breeding procedure. The Ensemble Transform Kalman Filter (ETKF) method is a natural extension of the concept of breeding and, as shown by Wang and Bishop, can be used to generate ensemble perturbations that can potentially ameliorate these shortcomings. In the present paper, a spherical simplex 10member ETKF ensemble, using the actual distribution and error characteristics of realtime observations and an innovationbased inflation, is tested and compared with a 5pair breeding ensemble in an operational environment. The experimental results indicate only minor differences between the performances of the operational breeding and the experimental ETKF ensemble and only minor differences to Wang and Bishop’s earlier comparison studies. As for the ETKF method, the initial perturbation variance is found to respond to temporal changes in the observational network in the North Pacific. In other regions, however, 10 ETKF perturbations do not appear to be enough to distinguish spatial variations in observational network density. As expected, the whitening effect of the ETKF together with the use of the simplex algorithm that centres a set of quasiorthogonal perturbations around the best analysis field leads to a
On the Sensitivity equations of fourdimensional variational (4DVar) data assimilation
 Mon. Weather Rev
, 2008
"... The equations of the forecast sensitivity to observations and to the background estimate in a fourdimensional variational data assimilation system (4DVar DAS) are derived from the firstorder optimality condition in unconstrained minimization. Estimation of the impact of uncertainties in the speci ..."
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Cited by 9 (2 self)
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The equations of the forecast sensitivity to observations and to the background estimate in a fourdimensional variational data assimilation system (4DVar DAS) are derived from the firstorder optimality condition in unconstrained minimization. Estimation of the impact of uncertainties in the specification of the error statistics is considered by evaluating the sensitivity to the observation and background error covariance matrices. The information provided by the error covariance sensitivity analysis is used to identify the input components for which improved estimates of the statistical properties of the errors are of most benefit to the analysis and forecast. A close relationship is established between the sensitivities within each input pair data/error covariance such that once the observation and background sensitivities are available the evaluation of the sensitivity to the specification of the corresponding error statistics requires little additional computational effort. The relevance of the 4DVar sensitivity equations to assess the data impact in practical applications is discussed. Computational issues are addressed and idealized 4DVar experiments are set up with a finitevolume shallowwater model to illustrate the theoretical concepts. Timedependent observation sensitivity and potential applications to improve the model forecast are presented. Guidance provided by the sensitivity fields is used to adjust a 4DVar DAS to achieve forecast error reduction through assimilation of supplementary data and through an accurate specification of a few of the background error variances. 1.
Controlling instabilities along a 3DVar analysis cycle by assimilating in the unstable subspace: a comparison with the EnKF
, 2008
"... A hybrid scheme obtained by combining 3DVar with the Assimilation in the Unstable Subspace (3DVarAUS) is tested in a QG model, under perfect model conditions, with a fixed observational network, with and without observational noise. The AUS scheme, originally formulated to assimilate adaptive obser ..."
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Cited by 6 (5 self)
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A hybrid scheme obtained by combining 3DVar with the Assimilation in the Unstable Subspace (3DVarAUS) is tested in a QG model, under perfect model conditions, with a fixed observational network, with and without observational noise. The AUS scheme, originally formulated to assimilate adaptive observations, is used here to assimilate the fixed observations that are found in the region of local maxima of BDAS vectors (Bred vectors subject to assimilation), while the remaining observations are assimilated by 3DVar. The performance of the hybrid scheme is compared with that of 3DVar and of an EnKF. The improvement gained by 3DVarAUS and the EnKF with respect to 3DVar alone is similar in the present model and observational configuration, while 3DVarAUS outperforms the EnKF during the forecast stage. The 3DVarAUS 1 algorithm is easy to implement and the results obtained in the idealized conditions of this study encourage further investigation toward an implementation in more realistic contexts.
2006: Verification region selection and data assimilation for adaptive sampling
"... Adaptive or targeted observations supplement routine observations at a prespecified targeting time. Adaptive observation locations are selected to supplement routine observations in an attempt to minimize the forecast error variance of a future target forecast within some predefined verification re ..."
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Cited by 4 (2 self)
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Adaptive or targeted observations supplement routine observations at a prespecified targeting time. Adaptive observation locations are selected to supplement routine observations in an attempt to minimize the forecast error variance of a future target forecast within some predefined verification region (VR) at some predefined verification time. Ideally, the VR is placed in a location where unusually large forecast errors are likely. Here, we compare three methods of selecting VRs. A climatological method based on seasonal averages of forecast errors. An unconditioned method based on verification time ensemble spread and a conditioned method based on an Ensemble Transform Kalman Filter (ETKF) estimate of forecast error variance given the routine observations to be taken at the targeting time. To test the effectiveness of the three approaches, Observation System Simulation Experiments (OSSEs) on a chaotic barotropic flow were performed using an imperfect model. To test the sensitivity of our results to the type of forecast error covariance model used in the data assimilation (DA) scheme, two types of DA schemes were tested: An isotropic DA scheme and a hybrid DA scheme. For isotropic DA, correlations between vorticity forecast errors at any two points were solely a function of the distance between the points. For hybrid DA, the
A Comparison of the Hybrid and EnSRF Analysis Schemes in the Presence of Model Errors due to Unresolved Scales
, 2009
"... A hybrid analysis scheme is compared with an ensemble square root filter (EnSRF) analysis scheme in the presence of model errors as a followup to a previous perfectmodel comparison. In the hybrid scheme, the ensemble perturbations are updated by the ensemble transform Kalman filter (ETKF) and the ..."
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Cited by 3 (0 self)
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A hybrid analysis scheme is compared with an ensemble square root filter (EnSRF) analysis scheme in the presence of model errors as a followup to a previous perfectmodel comparison. In the hybrid scheme, the ensemble perturbations are updated by the ensemble transform Kalman filter (ETKF) and the ensemble mean is updated with a hybrid ensemble and static backgrounderror covariance. The experiments were conducted with a twolayer primitive equation model. The true state was a T127 simulation. Data assimilation experiments were conducted at T31 resolution (3168 complex spectral coefficients), assimilating imperfect observations drawn from the T127 nature run. By design, the magnitude of the truncation error was large, which provided a test on the ability of both schemes to deal with model error. Additive noise was used to parameterize model errors in the background ensemble for both schemes. In the first set of experiments, additive noise was drawn from a large inventory of historical forecast errors; in the second set of experiments, additive noise was drawn from a large inventory of differences between forecasts and analyses. The static covariance was computed correspondingly from the two inventories. The hybrid analysis was statistically significantly more accurate than the EnSRF analysis. The improvement of the hybrid over the EnSRF was smaller when differences of forecasts and analyses were used to form the random noise and the static covariance. The EnSRF analysis was more sensitive to the size of the ensemble than the hybrid. A series of tests was conducted to understand why the EnSRF performed worse than the hybrid. It was shown that the inferior performance of the EnSRF was likely due to the sampling error in the estimation of the modelerror covariance in the mean update and the lessbalanced EnSRF initial conditions resulting from the extra localizations used in the EnSRF. 1.